2-ring

For other uses, see Two rings (disambiguation).

In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring. But the point is that one can also consider the situation in which an element of R comes with a "nontrivial automorphism".[1]

This line of generalization of a ring eventually leads to the notion of an En-ring.

## See also

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* Categorification
* Higher-dimensional algebra
* Lie n-algebra

## Further reading

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* John Baez, 2-Rigs in Topology and Representation Theory

## References

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1. ^ Lurie, J. (2004). "V: Structured Spaces". Derived Algebraic Geometry (Thesis).

* Laplaza, M. (1972). "Coherence for distributivity". Coherence in categories. Lecture Notes in Mathematics. Vol. 281\. Springer-Verlag. pp. 29–65\. ISBN 9783540379584.

## External links

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* http://ncatlab.org/nlab/show/2-rig

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Category theory

Key concepts

| Key concepts|

* Category
* Abelian
* Additive
* Concrete
* Pre-abelian
* Preadditive
* Bicategory
* Adjoint functors
* CCC
* Commutative diagram
* End
* Exponential
* Functor
* Kan extension
* Morphism
* Natural transformation
* Universal property
* Yoneda lemma

|

Universal constructions|

| Limits|

* Terminal objects
* Products
* Equalizers
* Kernels
* Pullbacks
* Inverse limit

|

Colimits|

* Initial objects
* Coproducts
* Coequalizers
* Cokernels and quotients
* Pushout
* Direct limit

Algebraic categories|

* Sets
* Relations
* Magmas
* Groups
* Abelian groups
* Rings (Fields)
* Modules (Vector spaces)

Constructions on categories|

* Free category
* Functor category
* Kleisli category
* Opposite category
* Quotient category
* Product category
* Comma category
* Subcategory

Higher category theory

| Key concepts|

* Categorification
* Enriched category
* Higher-dimensional algebra
* Homotopy hypothesis
* Model category
* Simplex category
* String diagram
* Topos

|

n-categories|

| Weak n-categories|

* Bicategory (pseudofunctor)
* Tricategory
* Tetracategory
* Kan complex
* ∞-groupoid
* ∞-topos

|

Strict n-categories|

* 2-category (2-functor)
* 3-category

Categorified concepts|

* 2-group
* 2-ring
* En-ring
* (Traced)(Symmetric) monoidal category
* n-group
* n-monoid

* Category
* Outline
* Glossary

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